Convergence Theorems for Asymptotically Nonexpansive Mappings in Banach Spaces
نویسندگان
چکیده
Let E be a uniformly convex Banach space, and let K be a nonempty convex closed subset which is also a nonexpansive retract of E. Let T : K → E be an asymptotically nonexpansive mapping with {kn} ⊂ [1,∞) such that P∞ n=1(kn − 1) < ∞ and let F (T ) be nonempty, where F (T ) denotes the fixed points set of T . Let {αn}, {βn}, {γn}, {αn}, {β′ n}, {γ′ n}, {α′′ n}, {β′′ n} and {γ′′ n} be real sequences in [0, 1] such that αn +βn + γn = αn +β ′ n + γ ′ n = α ′′ n + β ′′ n + γ ′′ n = 1 and ε ≤ αn, αn, α′′ n ≤ 1 − ε for all n ∈ N and some ε > 0, starting with arbitrary x1 ∈ K, define the sequence {xn} by setting ><>: zn = P (α′′ nT (PT ) n−1xn + β′′ nxn + γ ′′ nwn), yn = P (αnT (PT ) n−1zn + β′ nxn + γ ′ nvn), xn+1 = P (αnT (PT )n−1yn + βnxn + γnun), with the restrictions P∞ n=1 γn < ∞, P∞ n=1 γ ′ n < ∞ and P∞ n=1 γ ′′ n < ∞, where {wn}, {vn} and {un} are bounded sequences in K. (i) If E is real uniformly convex Banach space satisfying Opial′s condition, then weak convergence of {xn} to some p ∈ F (T ) is obtained; (ii) If T satisfies condition (A), then {xn} convergence strongly to some p ∈ F (T ).
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